Given two elliptic curves defined over \( \mathbb {Q}\)  with good ordinary reduction at prime \( p \)  whose residual Galois representations are congruent modulo reduction at \( p \) , Greenberg and Vatsal in [1]} showed that the vanishing of the \( \mu \) -invariant attached to the Pontryagin dual of the Selmer group over the cyclotomic \( \mathbb {Z}_p \) -extension of one curve implies the vanishing of the \( \mu \) -invariant for the other.
Their work uses an auxiliary Selmer group called nonprimitive dual Selmer group which has the same \( \mu \) -invariant as the Pontryagin dual of the Selmer group.
A similar approach has been taken in the study of elliptic curves with supersingular reduction for the signed Selmer groups.
The \( \mu \) -invariants of the Pontryagin dual of the signed Selmer groups are referred to as the signed \( \mu \) -invariants.
An analogue of Greenberg–Vatsal was proved by B. D. Kim in [2]} for the signed \( \mu \) -invariants over cyclotomic \( \mathbb {Z}_p \) -extension using nonprimitive dual Selmer groups.
Later, in [3]} the authors used a new technique to work with residual representations of elliptic curves and they improved upon the results of Kim in [2]}.
They defined a new Selmer group called the fine residual signed Selmer group
and studied its structure as an Iwasawa module, in particular, the vanishing of the signed \( \mu \) -invariants (cf. [3]}).
